Algebra and Geometry | Part IA, 2005

Let M\mathcal{M} be the group of Möbius transformations of C{}\mathbb{C} \cup\{\infty\} and let SL(2,C)S L(2, \mathbb{C}) be the group of all 2×22 \times 2 complex matrices with determinant 1 .

Show that the map θ:SL(2,C)M\theta: S L(2, \mathbb{C}) \rightarrow \mathcal{M} given by

θ(abcd)(z)=az+bcz+d\theta\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)(z)=\frac{a z+b}{c z+d}

is a surjective homomorphism. Find its kernel.

Show that every TMT \in \mathcal{M} not equal to the identity is conjugate to a Möbius map SS where either Sz=μzS z=\mu z with μ0,1\mu \neq 0,1, or Sz=z±1S z=z \pm 1. [You may use results about matrices in SL(2,C)S L(2, \mathbb{C}), provided they are clearly stated.]

Show that if TMT \in \mathcal{M}, then TT is the identity, or TT has one, or two, fixed points. Also show that if TMT \in \mathcal{M} has only one fixed point z0z_{0} then Tnzz0T^{n} z \rightarrow z_{0} as nn \rightarrow \infty for any zC{}.z \in \mathbb{C} \cup\{\infty\} .

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